Undefined Rational Expressions in Algebra

Questions and Answers

When is the expression $\frac{51x^2 + 24x}{17x}$ undefined?

x = 0 (correct)

x = -2

The expression is defined for all real numbers

x = -3/8

What values of x make the expression $\frac{24x^2 + 9x}{8x + 3}$ undefined?

x = -3/8 (correct)

x = 4

x = -4

x = 0

What is the simplified form of $\frac{105}{42}$?

1/2

2/3

7/13

5/2 (correct)

List the two values of x that make the expression $\frac{x^3 - 9x}{x^2 + 5x + 6}$ undefined.

x = -3, -2 Signup and view all the answers

What is the result of multiplying $\frac{12}{7}$ and $\frac{49}{36}$?

7/3 Signup and view all the answers

A rational expression is defined when its denominator is zero.

False Signup and view all the answers

Match the following rational expressions with when they are undefined:

$\frac{51x^2 + 24x}{17x}$ = x = 0 $\frac{24x^2 + 9x}{8x + 3}$ = x = -3/8 $\frac{x^2 + 6x + 8}{x^2 - 16}$ = x = -4, 4 Signup and view all the answers

A basic rule of fractions states that for any rational expression $\frac{ac}{bc}$, if _ and _, then $\frac{ac}{bc} = \frac{a}{b}$.

b ≠ 0, c ≠ 0 Signup and view all the answers

Study Notes

Undefined Rational Expressions

Rational numbers can be expressed as fractions of two integers; rational expressions are fractions of two polynomials.
Denominators in rational expressions cannot equal zero to avoid undefined values.
The domain of a rational expression includes all variable values except those that make the denominator zero.
To find undefined values, set the denominator to zero and solve for the variable.
Examples of expressions that are undefined:
- For ( \frac{51x^2 + 24x}{17x} ), undefined at ( x = 0 ).
- For ( \frac{24x^2 + 9x}{8x + 3} ), undefined at ( x = -\frac{3}{8} ).
- For ( \frac{x^3 - 9x}{x^2 + 5x + 6} ), undefined at ( x = -3, -2 ).
- For ( \frac{x^2 + 6x + 8}{x^2 - 16} ), undefined at ( x = -4, 4 ).
- The expression ( \frac{x^3 + 2x^2 + x + 2}{x^2 + 1} ) is defined for all real numbers as ( x^2 + 1 ) is never zero.

Simplifying Rational Expressions

Simplification follows the rule: if ( \frac{ac}{bc} ) (where ( b ≠ 0 ) and ( c ≠ 0 )), then ( \frac{ac}{bc} = \frac{a}{b} ).
To simplify, factor both the numerator and denominator, then divide out common factors.
Example simplifications:
- ( \frac{105}{42} = \frac{5}{2} ) after factoring.
- ( \frac{14}{21} = \frac{2}{3} ).
- ( \frac{21}{39} = \frac{7}{13} ).
- ( \frac{15x}{13x} = \frac{15}{13} ).
- ( \frac{2x + 6}{3x + 9} = \frac{2}{3} ).
- ( \frac{x^2 + 9x + 14}{x^2 - 4} = \frac{x + 7}{x - 2} ).
- ( \frac{x^2 + 2x - 3}{x^2 - 5x + 4} = \frac{x + 3}{x - 4} ).

Multiplying Rational Expressions

To multiply rational expressions, first factor the numerators and denominators completely.
Identify and reduce common factors before multiplying.
The Quotient Rule states ( \frac{x^a}{x^b} = x^{a-b} ) for any real numbers ( a ) and ( b ), under the condition ( x ≠ 0 ).
Example of multiplication:
- ( \left(\frac{12}{7}\right)\left(\frac{49}{36}\right) = \frac{7}{3} ) after factoring and reducing.

Description

This quiz focuses on understanding undefined rational expressions and the conditions under which they become undefined. You'll learn how to identify these expressions and simplify them effectively. Test your knowledge with various examples and practice solving for the variable that causes the denominator to equal zero.